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First Exit Time Models-Applications to datasets for various Countries

The best fit model is a combination of a model derived from the first exit time theory of a stochastic process with a Gompertzian correction for the ages from 15 to 30 years. 
For the best fit model go to the Health State Model webpage

Please download the latest version of the Regression model in Excel

Application of First Exit Time Models to Life Table Data of Various Countries 

The material included in this web page is a part from our project on modeling the Life Table Data sets for various countries by using Stochastic Modeling Techniques and especially the First Exit Time Distribution for a stochastic process which models the Health State of a Population. The advantages of using the stochastic theory are:

  • the modeling based on a function expressing the Health State of an individual as a stochastic process is according to the reality
  • the stochastic modeling makes feasible the realization of stochastic paths for the health state of an individual and then to find a distribution function for the health state over time for a population
  • by modeling the health state of a stochastic process we can define a barrier, usually a minimum level (in applications this level can be taken as zero)
  • the first exit time probability density function g(t) is estimated as a summation of the number of stochastic paths approaching for the first time the barrier
  • the use of the stochastic modeling technique provides powerful tools to demographers to have a Qhantitative estimate to what we call "Health State of a Population"

The first modeling approach based on the hitting time or first exit time theory for a stochastic process was developed by: Janssen, J. and Skiadas, C. H., Dynamic Modelling of Life-Table Data. Applied Stochastic Models and Data Analysis, vol. 11, No 1, 35-49, 1995. The paper can be downloaded at:http://onlinelibrary.wiley.com/doi/10.1002/asm.3150110106/abstract. The paper was also presented at a meeting in Brussels (1995) celebrating the 100 years of the Association Royale des Actuaires Belges. The model proposed in the first paper of 1995 is a 5 parameter first exit time model. The model was applied to the mortality data of Belgium and France and stochastic simulations where done along with the nonlinear estimation of the model's parameters.

The last five years we are working in simpler but yet quite powerful models to develop the distribution of the deaths per age in a large population or a country. The more successful models are presented and applied in the following. The nonlinear regression analysis technique used appears in this website along with easy to use Excel programs.

The data used are from the Human Mortality Database, HMD (www.mortality.org ) 

The models are the First Exit Time Models

The simpler model termed as "First Exit Time Model" is applied in recent years to life table data when the infant mortality is low.

This model is included in the paper proposed by: Skiadas, C. H. and Skiadas, C., A Modeling Approach to Life Table Data Sets, in Recent Advances in Stochastic Modeling and Data Analysis, World Scientific, 350 - 359, 2007 

 g(t)=kt-3/2exp(-(l-(bt)c)2/(2t))

The second model termed as "First Exit Time IM" models quite well the Infant Mortality and is here applied to the 1960 data for various countries when these data sets are available.

This model is included in the paper proposed by: Skiadas, C. and Skiadas, C. H., Development, Simulation and Application of First Exit Time Densities to Life Table Data, Communications in Statistics - Theory and Methods, Volume 39, Issue 3 January 2010, pages 444 - 451. This paper can be downloaded from: http://www.informaworld.com/smpp/content~db=all~content=a918382403~frm=titlelink  

The equation applied is: g(t)=kt-3/2(l+(c-1)(bt)c)exp(-(l-(bt)c)2/(2t))

FirstExitTimeModel-IM1.jpg

FirstExitTimeModel-IM2.jpg

FirstExitTimeModel-IM3.jpg

FirstExitTimeModel-IM4.jpg

The next Table summarizes the application results. The applications include the Nonlinear Regression Analysis Program in Excel and you may download it and apply more data from the Human Mortality Database at: www.mortality.org 

AUSTRALIA

First Exit Time Model-Australia-Females2007

First Exit Time Model-IM_Australia-Females-1960

LATVIA

First Exit Time Model-Latvia-Females-2007

First Exit Time Model-IM_Latvia-Females-1960

AUSTRIA

First Exit Time Model-Austria-Females-2008

First Exit Time Model-IM_Austria-Females-1960

LITHUANIA

First Exit Time Model-Lithuania-Females-2007

First Exit Time Model-IM_Lithuania-Females-1960

BELARUS

First Exit Time Model-Belarus-Females-2007

First Exit Time Model-IM_Belarus-Females-1960

LUXEMBURG

First Exit Time Model-Luxemburg-Females-2006

First Exit Time Model-IM_Luxemburg-Females-1960

BELGIUM

First Exit Time Model-Belgium-Females-2007

First Exit Time Model-IM_Belgium-Females-1960

NETHERLANDS

First Exit Time Model-Netherlands-Females-2006

First Exit Time Model-IM_Netherlands-Females-1960

BULGARIA

First Exit Time Model-Bulgaria-Females-2007

First Exit Time Model-IM_Bulgaria-Females-1960

NEW ZEALAND

First Exit Time Model-New Zealand-Females-2003

First Exit Time Model-IM_New Zealand-Females-1960

CANADA

First Exit Time Model-Canada-Females-2006

First Exit Time Model-IM_Canada-Females-1960

NORWAY

First Exit Time Model-Norway-Females-2007

First Exit Time Model-IM_Norway-Females-1960

CHILE

First Exit Time Model-Chile-Females-2005

First Exit Time Model-IM_Chile-Females-1992

POLAND

First Exit Time Model-Poland-Females-2006

First Exit Time Model-IM_Poland-Females-1960

CZECH REPUBLIC

First Exit Time Model-Czech Republic-Females-2008

First Exit Time Model-IM_Czech Republic-Females-1960

PORTUGAL

First Exit Time Model-Portugal-Females-2007

First Exit Time Model-IM_Portugal-Females-1960

DENMARK

First Exit Time Model-Denmark-Females-2007

First Exit Time Model-IM_Denmark-Females-1960

RUSSIA

First Exit Time Model-Russia-Females-2008

First Exit Time Model-IM_Russia-Females-1960

ESTONIA

First Exit Time Model-Estonia-Females-2007

First Exit Time Model-IM_Estonia-Females-1960

SLOVAKIA

First Exit Time Model-Slovakia-Females-2008

First Exit Time Model-IM_Slovakia-Females-1960

FINLAND

First Exit Time Model-Finland-Females-2008

First Exit Time Model-IM_Finland-Females-1960

SLOVENIA

First Exit Time Model-Slovenia-Females-2006

First Exit Time Model-IM_Slovenia-Females-1983

FRANCE

First Exit Time Model-France-Females-2007

First Exit Time Model-IM_France-Females-1960

SPAIN

First Exit Time Model-Spain-Females-2006

First Exit Time Model-IM_Spain-Females-1960

GERMANY

First Exit Time Model-Germany-Females-2008

First Exit Time Model-IM_Germany-Females-1990

SWEDEN

First Exit Time Model-Sweden-Females-2007

First Exit Time Model-IM_Sweden-Females-1960

GREECE

First Exit Time Model-Greece-Females-2000

First Exit Time Model-IM_Greece-Females-1982

SWITZERLAND

First Exit Time Model-Switzerland-Females-2007

First Exit Time Model-IM_Switzerland-Females-1960

HUNGARY

First Exit Time Model-Hungary-Females-2006

First Exit Time Model-IM_Hungary-Females-1960

TAIWAN

First Exit Time Model-Taiwan-Females-2008

First Exit Time Model-IM_Taiwan-Females-1960

ICELAND

First Exit Time Model-Iceland-Females-2007

First Exit Time Model-IM_Iceland-Females-1960

UK

First Exit Time Model-UK-Females-2006

First Exit Time Model-IM_UK-Females-1960

IRELAND

First Exit Time Model-Ireland-Females-2006

First Exit Time Model-IM_Ireland-Females-1960

UKRAINE

First Exit Time Model-Ukraine-Females-2008

First Exit Time Model-IM_Ukraine-Females-1960

ISRAEL

First Exit Time Model-Israel-Females-2007

First Exit Time Model-IM_Israel-Females-1983

USA

First Exit Time Model-USA-Females-2006

First Exit Time Model-IM_USA-Females-1960

ITALY

First Exit Time Model-Italy-Females-2006

First Exit Time Model-IM_Italy-Females-1960

   

JAPAN

First Exit Time Model-Japan-Females-2008

First Exit Time Model-IM_Japan-Females-1960

   

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